ISSN:
1572-8897
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Chemie und Pharmazie
,
Mathematik
Notizen:
Abstract The low-rank perturbation (LRP) method solves the perturbed eigenvalue equation (B +V)Ψ k = ɛ k (C +P)Ψ k , where the eigenvalues and the eigenstates of the related unperturbed eigenvalue equationBΦ i = λ i CΦ i are known. The method is designed for arbitraryn-by-n matricesB, V, C, andP, with the only restriction that the eigenstates Φ i of the unperturbed equation should form a complete set. We consider here a real LRP problem where all matrices are Hermitian, and where in addition matricesC and (C +P) are positive definite. These conditions guarantee reality of the eigenvalues ɛ k and λ i . In the original formulation of the LRP method, each eigenvalue ɛ k is obtained iteratively, starting from some approximate eigenvalue ɛ k ′ . If this approximate eigenvalue is not well chosen, the iteration may sometimes diverge. It is shown that in the case of a real LRP problem, this danger can be completely eliminated. If the rank ρ of the generalized perturbation {V, P} is “small” with respect ton, then one can easily bracket and hence locate to any desirable accuracy the eigenvalues ɛ k (k = 1, ...,n) of the perturbed equation. The calculation of alln eigenvalues requiresO(ρ2 n 2) operations. In addition, if the perturbation (V, P) is local with the localizabilityl ≈p, then onlyO(ρ2 n) operations are required for a derivation of a single eigenvalue.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF01172930
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